the development of utility theory stigler-1950

The Development of Utility Theory. IAuthor(s): George J. StiglerSource: The Journal of Political Economy, Vol. 58, No. 4 (Aug., 1950), pp. 307-327Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/1828885Accessed: 01/04/2010 19:46

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THE DEVELOPMENT OF UTILITY THEORY. I

GEORGE J. STIGLER Columbia University

But I have planted the tree of utility. I have planted it deep, and spread it wide.-BENTHAM.

T HE history of economic thought can be studied with many pur- poses. One may trace the effects

of contemporary economic and social conditions on economic theory or- rather more bravely-the effects of economic theories on economic and social developments. One may study the history to find the original discoverers of theories, spurred on by the dream of new Cantillons; or one may compare the economics of the great economists with that of the rank and file, as a contribution to the structure and process of intellectual change. Or one may, and most often does, simply set forth the major steps in the development of a branch of economic theory, hoping that it can be justified by its contribution to the un- derstanding of modern economics. This history of utility theory is offered primarily with this last purpose, although in the final section I review the history to answer the question, "Why do economists change their theories?"

The scope of this study is limited in several respects. First, it covers primarily the period from Smith to Slutsky, that is, from I776 to I9I5. Second, the study is limited to certain important topics and to the treatment of these topics by economists of the first rank. The application of utility theory to welfare economics is the most important topic omitted. An estimate of the part played by utility theory in forming economists' views of desirable social policy

is too large a task, in the complexity of issues and volume of literature involved, to be treated incidentally. The omission is justified by the fact that most economists of the period used utility theory primarily to explain economic behavior (particularly demand behavior) and only secondarily (when at all) to amend or justify economic policy.'

I. THE CLASSICAL BACKGROUND

ADAM SMITH

Drawing upon a long line of predecessors, Smith gave to his immediate suc- cessors, and they uncritically accepted, the distinction between value in use and value in exchange:

The word VALUE, it is to be observed, has two different meanings, and sometimes expresses the utility of some particular object, and sometimes the power of purchasing other goods which the possession of that object conveys. The one may be called "value in use"; the other, "value in exchange." The things which have the greatest value in use have frequently little or no value in exchange; and on the contrary, those which have the greatest value in exchange have frequently little or no value in use. Nothing is more useful than water: but it will purchase scarce any thing; scarce any thing can be had in exchange for it. A diamond, on

1I have also omitted consideration of the criticisms raised by the antitheoretical writers, who played no constructive part in the development of the theory. For a discussion of some of their views see J. Viner, "The Utility Theory and Its Critics," Journal of Political Economy, XXXIII (I925),

369-87. I wish to acknowledge the helpful suggestions of

Arthur F. Burns, Milton Friedman, and Paul A. Samuelson.

307

308 GEORGE J. STIGLER

the contrary, has scarce any value in use; but a very great quantity of other goods may frequently be had in exchange for it.2

The fame of this passage rivals its am- biguity.

The paradox-that value in exchange may exceed or fall short of value in use -was, strictly speaking, a meaningless statement, for Smith had no basis (i.e., no concept of marginal utility of income or marginal price of utility) on which he could compare such heterogeneous quantities. On any reasonable interpretation, moreover, Smith's statement that value in use could be less than value in exchange was clearly a moral judgment, not shared by the possessors of diamonds. To avoid the incompara- bility of money and utility, one may interpret Smith to mean that the ratio of values of two commodities is not equal to the ratio of their total utilities.' On such a reading, Smith's statement de- serves neither criticism nor quotation.

2 The Wealth of Nations (New York: Modern Library, I937), p. 28.

s Or, alternatively, that the ratio of the prices of two commodities is not equal to the ratio of their total utilities; but this also requires an illegitimate selection of units: The price of what quantity of diamonds is to be compared with the price of one gallon of water? Smith makes such illegitimate statements; for example, "The whole quantity of a cheap commodity brought to market, is commonly not only greater, but of greater value, than the whole quantity of a dear one. The whole quantity of bread annually brought to market, is not only greater, but of greater value than the whole quantity of butcher's-meat; the whole quantity of butcher's meat, than the whole quantity of poultry; and the whole quantity of poultry, than the whole quantity of wild fowl. There are so many more purchases for the cheap than for the dear commodity, that, not only a greater quantity of it, but a greater value, can commonly be disposed of" (ibid., p. 212; see also p. 838).

Nevertheless, this statement can be reformulated into a meaningful and interesting hypothesis: Order commodities by the income class of consumers, using the proportion of families in the income class that purchase the commodity as the basis for choos- ing the income class. Then does aggregate value of output fall as income class rises?

This passage is not Smith's title to recognition in our history of utility. His role is different: it is to show that demand functions, as a set of empirical relationships, were already an established part of economic analysis. The negatively sloping demand curve was already axiomatic; for example, "A competition will immediately begin among [the buyers when an abnormally small supply is available], and the market price will rise more or less above the natural price."4 The effect of income on consumption was not ignored:

The proportion of the expence of house-rent to the whole expence of living, is different in the different degrees of fortune. It is perhaps highest in the highest degree, and it diminishes gradually through the inferior degrees, so as in general to be lowest in the lowest degree. The necessaries of life occasion the great expence of the poor. They find it difficult to get food, and the greater part of their little revenue is spent in getting it. The luxuries and vanities of life occasion the principal expence of the rich; and a magnificent house embellishes and sets off to the best advantage all the other luxuries and vanities which they possess. A tax upon house- rents, therefore, would in general fall heaviest upon the rich; and in this sort of inequality there would not, perhaps, be any thing very unreasonable.5

This type of demand analysis was continued and improved by Smith's succes- sors, but his example should suffice to remind us that a history of utility is not a history of demand theory.

BENTHAM

Jeremy Bentham brought the principle of utility (to be understood much more broadly than is customary in economics) to the forefront of discussion in

4Ibid., p. 56. Substitution is illustrated by the effects of a royal death on the prices of black and colored cloth (ibid., p. 59).

5Ibid., pp. 793-94. This is of course the opposite of modern budgetary findings, but near-contemporary budget studies seem to me indirectly to support Smith.

UTILITY THEORY 309

England at the beginning of the nine- teenth century. In his Introduction to the Principles of Morals and Legislation (I 789) he suggested the measurement of quantities of pleasure and pain (primarily for the purpose of constructing a more rational system of civil and crim- inal law). Four dimensions of pleasure and pain were distinguished for the individual: (i) intensity, (2) duration, (3) certainty, and (4) propinquity.6

The first two dimensions are clearly relevant to the measurement of a pleasure, but the latter two are better treated as two of the factors which influence an individual's response to a particular pleasure or pain.' Bentham did not give explicit directions for calculating a given pleasure and indeed devoted a long chapter (vi) to "Circumstances Influ- encing Sensibility," which listed no less than thirty-two circumstances (such as age, sex, education, and firmness of mind) that must be taken into account in carrying out such a calculation.

The theory was much elaborated with respect to economic applications in Traits de legislation (i802), a lucid synthesis of many manuscripts made by his disciple, Etienne Dumont.8 Bentham was particularly concerned with the problem of equality of income, and this

6 Op. cit., chap. iv. In addition, two further "dimensions" were added for the appraisal of the total satisfaction of an "act": the consumption of a loaf of bread might be the pleasure to which the first four dimensions refer; the theft of the loaf might be the act. These additional dimensions were fecundity and purity; respectively, the chance of one pleasure leading to another and the chance of a pleasure not being followed by a pain.

'As Bentham indicated elsewhere (see Works of Jeremy Bentham [Edinburgh: Tait, i843], I, 206;

III, 214).

8 The reliability of the presentation of Bentham's views has been attested by Elie Hal6vy, La Forma- tion du radicalism philosophique (Paris: Germer Bailliere, i9oi), Vol. I, Appendix I. Here the Hil- dreth translation of the Traites is used (London: TrUbner, I87I).

raised the question of comparisons of the utilities of persons who might differ in thirty-two circumstances:

It is to be observed in general, that in speaking of the effect of a portion of wealth upon happiness, abstraction is always to be made of the particular sensibility of individuals, and of the exterior circumstances in which they may be placed. Differences of character are inscru- table; and such is the diversity of circumstances, that they are never the same for two individuals. Unless we begin by dropping these two consid- erations, it will be impossible to announce any general proposition. But though each of these propositions may prove false or inexact in a given individual case, that will furnish no argument against their speculative truth and practical utility. It is enough for the justification of these propositions-Ist, If they approach nearer the truth than any others which can be substi- tuted for them; 2nd, If with less inconvenience than any others they can be made the basis of legislation

Thus, he achieved interpersonal comparisons, not by calculation, but by assumption, justified by the desirability (somehow determined) of its corollaries. This resort to a question-begging assumption was a fundamental failure of his project to provide a scientific basis for social policy: the scientific basis was being justified by the policies to which it led. In one of his manuscripts he argued that this assumption was merely an abbreviation and that the conclusions he deduced could be reached (more laboriously) without it,10 which is not in general true.

Theory of Legislation, p. I03.

1O" 'Tis in vain to talk of adding quantities which after the addition will continue distinct as they were before, one man's happiness will never be another man's happiness; a gain to one man is no gain to another: you might as well pretend to add 20 apples to 20 pears, which after you had done that could not be 40 of any one thing but 20 of each just as there was before. This addibility of the happiness of different subjects, however, when considered rigorously it may appear fictitious, is a postulatum without the allowance of which all political reasoning is at a stand: nor is it more

3IO GEORGE J. STIGLER

Having surmounted this obstacle no better than subsequent economists, Bentham proceeded to establish a set of propositions on the utility of income :11 ist. Each portion of wealth has a correspond-

ing portion of happiness. 2nd. Of two individuals with unequal fortunes,

he who has the most wealth has the most happiness.

3rd. The excess in happiness of the richer will not be so great as the excess of his wealth.12

Each of these propositions was elaborated, and the utility calculus was used to defend equality ("The nearer the actual proportion approaches to equality, the greater will be the total mass of happiness"), although equality was finally rejected in favor of security of property. As corollaries, gambling was utility-decreasing and insurance utility- increasing."3

fictitious than that of the equality of chances to reality, on which that whole branch of the Mathe- matics which is called the doctrine of chances is established. The fictitious form of speech (expression) in both cases, which, fictitious as it is, can give birth to no false consequences or conclusions, is adopted from a necessity which induces the like expedient in so many other instances, merely for the sake of abbreviation: as it would be endless to repeat in every passage where it was used, what it was it wanted to be rigorously true" (Halhvy, op. cit., III, 48i).

" Theory of Legislation, pp. 103 ff.; all statements italicized by Bentham.

12The use of marginal analysis was even more explicit in his Pannomicil Fragments:

"But the quantity of happiness will not go on increasing in anything near the same proportion as the quantity of wealth:-ten thousand times the quantity of wealth will not bring with it ten thousand times the quantity of happiness. It will even be matter of doubt whether ten thousand times the wealth will in general bring with it twice the happiness.

". . . the quantity of happiness produced by a particle of wealth (each particle being of the same magnitude) will be less and less at every particle; . . ." (Works, III, 229; see also IV, 541).

Theory of Legislation, pp. io6-7.

In a manuscript written about I782,

Bentham attempted to set forth more clearly the precise measurement of utility."4 We are given a definition of the unit of intensity:

The degree of intensity possessed by that pleasure which is the faintest of any that can be distinguished to be pleasure, may be represented by unity. Such a degree of intensity is in every day's experience: according as any pleasures are perceived to be more and more intense, they may be represented by higher and higher numbers: but there is no fixing upon any particular degree of intensity as being the highest of which a pleasure is susceptible.15

(This suggested measure will be discussed in connection with the Weber- Fechner literature.) Then, shifting ground, Bentham argues that, although utility does not increase as fast as income, for small changes the two move proportionately,'6 so we may measure pleasures through the prices they com- mand:

If then between two pleasures the one produced by the possession of money, the other not, a man had as lief enjoy the one as the other, such pleasures are to be reputed equal. But the pleasure produced by the possession of money, is as the quantity of money that produces it: money is therefore the measure of this pleasure. But the other pleasure is equal to this; the other pleasure therefore is as the money that produces this: therefore money is also the measure of that other pleasure.17

Unfortunately, this procedure is illegitimate; we cannot use an equality (or, more strictly, a constancy of the marginal utility of money) that holds for small changes to measure total utilities.'8 These suggestions are impor-

14 Lengthy extracts are given by Halvy, op. cit., Vol. I, Appendix II.

"Ibid., p. 398. "lbid., p. 408. " Ibid., p. 4IO.

18 Bentham appears to have recognized this diffi- culty when, in a passage following a discussion of

UTILITY THEORY 31I

tant chiefly in revealing Bentham's awareness of the crucial problems in his calculus and his ingenuity in attempting to solve them.19

Bentham had indeed planted the tree of utility. No reader could overlook the concept of utility as a numerical magnitude; and the implications for economic analysis were not obscure. But they were overlooked.

THE RICARDIANS

The economists of Bentham's time did not follow the approach he had opened. One may conjecture that this failure is due to the fact that Ricardo, who gave the economics of this period much of its slant and direction, was not a Benthamite. It is true that he was the friend of Bentham and the close friend of James Mill, Bentham's leading disciple. Yet there is no evidence that he was a devout utilitarian and much evidence that he was unphilosophical- essentially a pragmatic reformer.20

It is clear, in any event, that Ricardo did not apply the utility calculus to economics. He began his Principles with the quotation of Smith's distinction between value in use and value in exchange and ended the volume with the statement: "Value in use cannot be measured by any known standard; it is

diminishing marginal utility, he wrote: " [Intensity] is not susceptible of precise expression: it not being susceptible of measurement" (Codification Proposal [i822], in Works, IV, 542).

'9For more general discussions of Bentham see W. C. Mitchell, "Bentham's Felicific Calculus," in The Backward Art of Spending Money (New York: McGraw-Hill Book Co., I937); and J. Viner, "Ben- tham and J. S. Mill," American Economic Review, XXXIX (I949), 360-82.

20 See Bonar's Preface to Letters of Ricardo to Malthus (Oxford: Clarendon, i887).

differently estimated by different persons."'21 I should be content to notice that he left the theory of utility as highly developed as he found it-as much cannot be said for the theory of value-were it not for a remarkable interpretation of Marshall's:

Again, in a profound, though very incom- plete, discussion of the difference between "Value and Riches" he seems to be feeling his way towards the distinction between marginal and total utility. For by Riches he means total utility, and he seems to be always on the point of stating that value corresponds to the increment of riches which results from that part of the commodity which it is only just worth the while of purchasers to buy; and that when the supply runs short, whether temporarily in consequence of a passing accident, or permanently in consequence of an increase in cost of production, there is a rise in that marginal increment of riches which is measured by value, at the same time that there is a diminution in the aggregate riches, the total utility, derived from the commodity. Throughout the whole discussion he is trying to say, though (being igno- rant of the terse language of the differential calculus) he did not get hold of the right words in which to say it neatly, that marginal utility is raised and total utility is lessened by any check to supply.22

In the chapter (xx) referred to, Ri- cardo defines riches as "necessaries, conveniences, and amusements," and value, as usual, is measured by the amount of labor necessary to produce a commodity. The chapter is essentially an exercise in the paradoxes of this definition of value; for example, if the productivity of labor doubles, riches double, but value changes only if the number of laborers changes. We may properly identify "necessaries, conveniences, and amusements" with total

Principles of Political Economy and Taxation (Gonner ed.; London: Bell, I932), p. 420.

22 Principles of Economiics (8th ed.; London: Macmillan, 1920), p. 814.

3I 2 GEORGE J. STIGLER

utility; but what of marginal utility? Ricardo says that, if a person receives two sacks of corn where formerly he received one, "he gets, indeed double the quantity of riches-double the quantity of utility-double the quantity of what Adam Smith calls value in use."23 Hence he did not believe that marginal utility diminishes as quantity increases. He continued:

When I give 2,000 times more cloth for a pound of gold than I give for a pound of iron, does it prove that I attach 2,000 times more utility to gold than I do to iron? certainly not; it proves only as admitted by M. Say, that the cost of production of gold is 2,000 times greater than the cost of production of iron . . if utility were the measure of value, it is probable I should give more for the iron.24

The writer of this passage cannot be said to have been close to the notion of marginal utility. I cannot find a single sentence that gives support to Mar- shall's interpretation, and I think that it should be added to the list of ex- amples of his peculiar documentation and interpretation of predecessors.

Ricardo's influence was such that James Mill, the logical person to apply Bentham's system to economics, was content to present a rigid simplification of Ricardo's Principles ;25 and his son- whose formative work in economics, we must remember, came chiefly in the i82o's-did little more with utility.26 Only the French utilitarian, J. B. Say, attempted to give utility a substantial place in economic theory, and he was prevented from doing so effectively by his inability to arrive at a notion of marginal analysis. In order to support the

22 Principles, p. 265. 24 Ibid., pp. 267-68. ' In his Elements of Political Economy (3d ed.,

I827). 26 Principles of Political Economy (Ashley ed.;

New York: Longmans, Green, 1929), pp. 442-44, 804.

thesis that prices are proportional to utilities, he was driven to invent the metaphysical distinction between natural and social wealth:

One pays 2,000 times as much for a pound of gold as for a pound of iron. Here is how, on my theory, this phenomenon is explained. I assume with you that a pound of iron has the same utility as a pound of gold, although it is worth only one-two-thousandth as much. I say that there are in the iron I,999 degrees of utility that nature has given us without charge, and i degree that we create by work, at an expense that we will assume only if a consumer is willing to reimburse us; hence the pound of iron has 2,000 degrees of utility. The gold also has 2,000 degrees of utility (on your assumption), which however can be obtained only on exacting terms, that is to say, . . . by expenses of 2,000. The i,999 degrees of utility for which we do not pay when we consume iron are part of our natural wealth.... The single degree of utility which must be paid for is part of our social wealth.27

II. THE UNSUCCESSFUL DISCOVERERS

The principle that equal increments of utility-producing means (such as income or bread) yield diminishing increments of utility is a commonplace. The first statement in print of a commonplace is adventitious; it is of no importance in the development of economics, and it confers no intellectual stature on its author. The statement acquires interest only when it is logi- cally developed or explicitly applied to economic problems, and it acquires importance only when a considerable number of economists are persuaded to in- corporate it into their analyses. Inter- est and importance are of course mat- ters of degree.

Some economists gave clear state- 'Letter to Ricardo, July i9, 1821, in Melanges

et correspondance (Paris: Chamerot, I833), pp. ii6-i7, 287-89; cf. also Treatise on Political Economy (Boston: Wells & Lilly, 1824), Book II, chap. i, and Cours corn plet d'6conomnie politique (Paris: Guillaumin, 1840), I, 65-66, 71-72.

UTILITY THEORY 313

ments of the principle of diminishing marginal utility but did not apply it to economic problems; they include Lloyd (1833), Senior (I836), Jennings (i855), and Hearn (i864).28 Others applied utility theory to economic events without explicitly developing the principle of diminishing marginal utility: A. Walras (I83I) and Long- field (I834), for example.29 At least two economists-in addition to Ben- tham-elaborated the principle or applied it to economic problems but failed to persuade other economists of its use- fulness.30 Their theories will be sum- marized briefly.

DUPUIT (1844)

Jules Dupuit, a distinguished en- gineer, was led to the marginal utility theory by his attempt to construct a theory of prices that maximize utility.31 He distinguished total and marginal utility with great clarity and discov- ered "une espece de benefice" that we now call consumers' surplus. It was de-

28 W. F. Lloyd, "The Notion of Value," reprinted in Economic History, Economic Journal Supple- ment, May, 1927, pp. 170-83; N. W. Senior, Polit- ical Economny (New York: Farrar & Rinehart, 1939), pp. I I-I 2; R. Jennings, Natural Elements of Political Econom1iy (London: Longman, Brown, Green & Longmans, 1855), pp. 98-99, 19, 233 n.; W. E. Hearn, Plutology (London: Macmillan, i864), p. 17. Lloyd, the third occupant of the Drummond chair in political economy at Oxford, gave much the most elaborate statement of the principle. Instead of applying it to contemporary economic problems, however, he emphasized the fact that marginal utility is not the same thing as exchange value and applied the theory to Robinson Crusoe to show this.

29A. Walras, De la nature de la richesse et de l'origine de la valeur (Paris: Alcan, 1938), esp. chap. xi; M. Longfield, Lectures on Political Econ- oiny ("London School Reprints" [London, 1931]), pp. 2 7-28, 45-46, i i i ff.

30 Daniel Bernoulli's much earlier discovery will be treated later.

31 His chief essays (published in I844 and 1849) are reprinted in De l'utilitW et de sa mesure (Torino: La Riforma Sociale, 1934).

fined as the excess of total utility over marginal utility times the number of units of the commodity, but it was ac- tually taken to be the area under the demand curve minus the expenditures on the commodity (i.e., Marshall's measure without his restrictions).32

Armed with this concept, he investi- gated the optimum toll on a bridge.

P

P q n

C r r' N

FIG. I

His analysis was as follows. Let NP be the demand (and marginal utility) curve, Op the price (Fig. i). Then OrnP is the absolute utility consumers obtain from the use of the bridge, and pnP is the relative utility. If the toll is reduced by pp', there is a net gain of consumer utility of qnn' (equal to the area under the demand curve between r and r' minus the expenditure rr'n'q).

Dupuit's general conclusion is: "The utility of a means of communication, and in general of any product, is at a

32 Dupuit's instruction for measuring utility re- veals the tacit identification of utility and demand curves: "Assume that all the like commodities whose general utility one wishes to determine are subjected to a tax which is increased by small steps. At each increase, a certain quantity of the commodity will no longer be purchased. The utility of this quantity in terms of money will be the quantity multiplied by the tax. By increasing the tax until all purchases cease, and adding the partial products, one will obtain the total utility of the commodity" (ibid., p. 50; also p. I8O).


maximum when the toll or the price is zero."33 This is little more than a tautol- ogy, and Dupuit did not draw the further and illegitimate conclusion that the optimum toll rate is zero:

It will not be our conclusion [that tolls should be small or zero], when we treat of tariffs; but we hope to have demonstrated that [tariff rates] must be studied, combined on rational principles to produce simultaneously the greatest possible utility and a revenue which will repay the expense of maintenance and the interest on the capital investment.34

We see that he was not afraid of interpersonal comparisons of utility, and in fact he argued that the effects of price changes on the distribution of income must be ignored because they were merely transfers.35

Dupuit could not reach a complete theory of optimum prices because he did not devise a coherent theory of cost.36 One is impressed by the narrowness of

" Ibid., p. I6i. I have transposed the axes of Dupuit's diagram.

4 Ibid., p. 5i. Elsewhere he says that the ideal toll would be one proportional to the consumers' total utility, but this is impracticable because of "l'improbit6 universelle" (ibid., p. 14) ; and the effects of alternative methods of financing public works (e.g., the incidence of taxes) must be studied before a practical recommendation can be made (ibid., p. i6i). Multiple price systems were also considered (ibid., pp. 64-65, 140 ff.).

" Ibid., p. 52.

16This is illustrated by the following quotation, in which price fluctuations are treated as exercises of arbitrary power:

"In order that there be an increase or decrease in utility, it is necessary that there be a decrease or increase in [a commodity's] cost of production- there being no change in its quality. When there are only variations in market price [prix venal], the consumer gains what the producer loses, or conversely. Thus, when an article costing 20 francs to produce is sold for 5o francs, as a result of a monopoly or concession, the producer deprives every buyer of 30 francs of utility. If some cir- cumstance forces him to lower his price by io francs, his income diminishes by io francs per unit and that of each buyer increases by io francs. There is a cancellation; no utility is produced" (ibid., pp. 52-53).

his vision; the explicit formulation of the concept of consumer surplus is elegant, but there is no intuition of the difficulties in the concept, nor is there an attempt to construct the larger theoreti- cal framework necessary to solve his problem.

GOSSEN (I854)

Heinrich Gossen is one of the most tragic figures in the history of economics. He was a profound, original, and untrained thinker who hid his thoughts behind painfully complex arithmetical and algebraic exercises.37 He displayed every trait of the crank,38 excepting only one: history has so far believed that he was right. Only a few distinctive features of his work will be commented upon.

First, Gossen's discussion of the laws of satisfaction is concerned only with individual acts of consumption, such as the eating of slices of bread.39 Corre- spondingly, in his early diagrams marginal utility is a function of time (duration of the act of consumption), and only after a considerable elaboration of this approach does he take quantity of a (perishable) commodity as proportional to duration of consumption.40

3 Only a person who has labored through the volume can savor the magnificent understatement of Edgeworth: "He may seem somewhat deficient in the quality of mathematical elegance" ("Gossen," Palgrave's Dictionary of Political Economy [London: Macmillan, 1923], II, 232).

38 His Entwickelung der Gesetze des vmenschlichen Verkehrs (3d ed.; Berlin: Prager, 1927), which is not encumbered with chapters, begins with the famous sentences: "On the following pages I sub- mit to public judgment the result of 20 years of meditation. What a Copernicus succeeded in ex- plaining of the relationships of worlds in space, that I believe I have performed for the explanation of the relationships of men on earth."

"9 For a good summary see M. Pantaleoni, Pure Economics (London: Macmillan, 1898), pp. 28 ff.

40Entwickelung, p. 29; his treatment of durable goods is not sound (see pp. 25, 29-30).

UTILITY THEORY 315

Yet he does not attempt to work out a theory of the temporal pattern of consumption, and this portion of his theory seems misdirected.

Second, he presents a theory of the marginal disutility of labor that is com- pletely symmetrical with that of the marginal utility of consumer goods. Gossen's curve of the marginal disutility of income is essentially identical with that which Jevons made famous: the early hours of work yield utility, but, as the duration of labor increases, the marginal utility diminishes to zero and then to negative values.4" He defines the condition of maximum utility as that in which the marginal utility of a unit of product is numerically equal to the marginal disutility of the labor necessary to produce a unit of product.42

Third, Gossen was the first writer to formulate explicitly what I shall call the fundamental principle of marginal utility theory:

A person maximizes his utility when he dis- tributes his available money among the various goods so that he obtains the same amount of satisfaction from the last unit of money (Geld- atom) spent upon each commodity.43

We may translate this statement into semisymbolic form:

MU1 MU2 MU3 P1 P2 P3 '

where MUi represents the marginal utility of the ith commodity and pi its price. (We shall adhere to the notation: xi is the quantity of commodity Xi, pi is its price, MUi is its marginal utility, and R is money income.) This equation marked a long step forward in the development of the relationship be-

4lIbid., p. 36. 42 Ibid., p. 45.

43Ibid., pp. 93-94.

tween utility and demand curves. Finally, Gossen's views on the meas-

urability of utility are vague but tanta- lizing:

We can conceive of the magnitudes of various pleasures only by comparing them with one another, as, indeed, we must also do in measuring other objects. We can measure the magnitudes of various areas only by taking a particular area as the unit of measurement, or the weights of different bodies only by taking a particular weight as the unit. Similarly, we must fix on one pleasure as our unit, and hence an indefiniteness remains in the measurement of a pleasure. It is a matter of indifference which pleasure we choose as the unit. Perhaps the consequences will be most convenient if we choose the pleasure from the commodity which we use as money.44

He did not notice that there might be no unit of utility comparable with that of area or weight; and it is probably going too far to read into this passage the later position that it is sufficient to deal with the ratios of marginal utilities.

III. THE BEGINNINGS OF THE

MODERN THEORY

The utility theory finally began to win a place in generally accepted economics in the i8 70's, under the triple auspices of Jevons, Menger, and Wal- ras. Independently these economists arrived at positions similar in the main and sometimes in detail.45 I shall compare their treatments of certain basic

4Ibid., p. I23.

"Marshall was a contemporary discoverer of the theory but did not publish it until later (Memori- als of Alfred Marshall [London: Macmillan, 1925],

p. 2 2). J. B. Clark was a somewhat later discoverer and never developed the theory to a level comparable with the best contemporary European analysis. He became preoccupied with a neglected problem to which he could not find a useful solution: how to apply marginal analysis to variations in the quality of goods (see The Philosophy of Wealth [Boston: Ginn & Co., i892], Preface and p. 76 n.; Distribution of Wealth [New York: Mac- millan, 1931], chaps. xiv-xvi).

3i6 GEORGE J. STIGLER

problems of the theory, and henceforth our organization will be by subject.

A. CRITICISM OF RECEIVED DOCTRINE

Each of these founders of utility theory criticized the Ricardian theory of value, but for each this was an incidental and minor point; they deemed the positive merits of the utility theory a sufficient basis for acceptance. Thus, only after completing the presentation of his utility theory did Jevons point out the deficiencies in Ricardo's labor value theory. These deficiencies were three: (i) Ricardo required a special theory for commodities with fixed supplies, such as rare statues. This proved that labor cost is not essential to value.

,(2) Large labor costs will not confer high value on a commodity if the future demand is erroneously forecast; "in commerce bygones are for ever bygones."46 (3) Labor is heterogeneous, and the various types of labor can be compared only through the values of their products.47 On the other hand, the cost of production theory of value fits in nicely as a special case of the utility theory, for it explains the relative quantities of commodities that will be supplied.48

Menger and Walras took fundamen- tally the same position. The former also gave the first two criticisms listed above and, in addition, made a parallel criticism to the Ricardian rent theory: if the value of land did not depend upon labor cost, this demonstrated a serious lack of generality in the classical theory of value.49 Walras repeated the criti-

46 Theory of Political Economty (4th ed.; London: Macmillan, I91I), p. I64.

47Ibid., p. i66. 48Ibid., p. i65.

4 Grundsitze der Volkswirtschaftslehre (Vienna: BraumUller, i870), pp. 69, I20-21, 144-45.

cism that the classical theory lacked generality, emphasized the reciprocal effects of prices of products and of productive services on one another, and denied the existence of the class of commodities whose supplies could be infinitely increased, on the overly literal ground that no productive resource was available in infinite quantity.'

The task of elaborating and ex- pounding the theory, and of exaggerat- ing its merits and understating the use- fulness of the classical theory-the in- evitable accompaniments of intellectual innovations-fell largely to disciples, in particular Wieser and Bbhm-Bawerk. These men did not improve on the substance of the theory-in fact, it dete- riorated in their hands-so we shall pass them by.51

B. TIHE EXISTENCE AND MEASURABILITY

OF UTILITY

Without exception, the founders accepted the existence of utility as a fact of common experience, congruent with the most casual introspection. Jevons was most explicit:

The science of Economics, however, is in some degree peculiar, owing to the fact . . . that its ultimate laws are known to us immediately by intuition, or, at any rate, they are furnished to us ready made by other mental or physical sciences.

... The theory here given may be described as the mechanics of utility and self-interest. Oversights may have been committed in tracing

Et clients d'cononmie politique pure (I926 ed.; Paris: Pichon & Durand-Auzias), Lecon 38. The first edition (Lausanne: Carbay, I874) does not differ materially in substance on the subjects discussed here.

" Wieser's paradox of value (that marginal utility times quantity may decrease when quantity increases) led to deep confusion (see Natural Value [New York: Stechert, 1930], Books I and II). Binhm-Bawerk's greatest polenmic is Grundzihge der Theorie des wirtschaftlichen Giiterwerts ("London School Reprints" [London, 1932]).

UTILITY THEORY 317

out its details, but in its main features this theory must be the true one. Its method is as sure and demonstrative as that of kinematics or statics, nay, almost as self-evident as are the elements of Euclid. ...52

I am inclined to interpret the silence of Menger and Walras on the existence of utility as indicative of an equally complete acceptance.

Menger glossed over the problem of measurability of utility. He represented marginal utilities by numbers and employed an equality of marginal utilities in various uses as the criterion of the optimum allocation of a good.53 1-is word for utility - Bedeutung-was surely intentionally neutral, but probably it was chosen for its nonethical flavor.54 Walras was equally vague; he simply assumed the existence of a unit of measure of intensity of utility and thereafter spoke of utility as an absolute magnitude.55

Jevons' attack on the problem of measurability was characteristically frank and confused. He denied that utility was measurable:

There is no unit of labour, or suffering, or enjoyment.

I have granted that we can hardly form the conception of a unit of pleasure or pain, so that the numerical expression of quantities of feeling seems to be out of question.56

Yet he seemed also to argue that one cannot be sure that utility is not measurable but only that it could not pres- ently be measured.57 He was somewhat more skeptical of the measurability of

52Op. cit., pp. iS and 2 1.

Op. cit., p. 98 n.

5 On one occasion he states that his numbers represent only relative utilities and that numbers such as 8o and 40 indicate only that the former (marginal) utility is twice as large as the latter ibidd., p. i63 n.).

AEliments, pp. 74, I02, I53.

Op. cit., pp. 7 and 12. "Ibid., pp. i-9.

utility in the first (i871) than in the second (I879) edition; for example, in the second edition he deleted the following passage:

I confess that it seems to me difficult even to imagine how such estimations [of utility] and summations can be made with any approach to accuracy. Greatly though I admire the clear and precise notions of Bentham, I know not where his numerical data are to be found.58

With gallant inconsistency, he proceeded to devise a way to measure utility. It employed the familiar measuring rod of money:

It is from the quantitative effects of the feel- ings that we must estimate their comparative amounts.

I never attempt to estimate the whole pleasure gained by purchasing a commodity; the theory merely expressed that, when a man has purchased enough, he would derive equal pleasure from the possession of a small quantity more as he would from the money price of it."

This position is elaborated ingeniously: Wie can construct a demand curve by observation (or possibly experiment), and then we can pass to the marginal utility curve by means of the equation,

MUrPi=MUi X

where MU, is the marginal utility of income.60

For the first approximation we may assume that the general utility of a person's income is not affected by the changes of price of the commodity. ...

The method of determining the function of utility explained above will hardly apply, however, to the main elements of expenditure. The price of bread, for instance, cannot be properly brought under the equation in question, because, when the price of bread rises much, the resources of poor persons are strained, money

58 Theory of Political Economny (ist ed.; London: Macmillan, I87I), p. I2.

9 Theory (4th ed.), pp. ii and I3. G Ibid., pp. 146 ff. (Our notation.)


becomes scarcer with them, and [MU,], the marginalj utility of money, rises.61

This procedure is so similar to Mar- shall's that we may defer comment until we discuss the latter's more elaborate version.

Unlike Walras and Menger, Jevons considered the question of the interpersonal comparison of utilities. He ex- pressly argued that this was impos- sible62 but made several such comparisons, as we shall notice later. Menger avoided the subject and did not engage in such comparisons; and Walras made only incidental interpersonal comparisons.63

C. UTILITY MAXIMIZATION AND THE

DEMAND CURVE

Menger simply ignored the relationship between utility and demand. He was content to set some demand prices (he worked always with discontinuous schedules) which somehow represented marginal utilities64 and proceeded to an elementary discussion of pricing under bilateral monopoly (the indeterminacy of which was recognized), duopoly (the complications of which were not recognized-a competitive solution was given), and competition (in which the absence of a theory of production had predictable effects).65

Jevons' attempt to construct a bridge between utility and demand was se- riously hampered, I suspect, by his inability to translate any but simple

61 Ibid., pp. 147 and I48.

62Ibid., p. I4.

63 See Etudes d'economnie politique applique (Lausanne: Rouge, i898), pp. 295 if.; Etudes d'economie sociale (Lausanne: Rouge, i896), pp. 209 ff.

64 "The value that a good has for an economizing individual is equal to the significance of that want- satisfaction" (op. cit., p. I20; also chap. v).

" Ibid., pp. I77 ff., 208-9.

thoughts into mathematics. His fundamental equation for the maximization of utility in exchanges was presented as a fait accompli:

MU1 pi M U2 P2

This equation was satisfactory for an individual confronted by fixed prices, but how to apply it to competitive mar- kets?

Jevons devised two concepts to reach the market analysis: the trading body and the law of indifference. A trading body was the large group of buyers or sellers of a commodity in a competitive market.6" The law of indifference was that there be only one price in a market.67

He proceeded in the following peculiar manner. Let the equation of exchange be applied to each trading body; for each group of competitive individuals the equation will determine the relationship between the quantity offered and the quantity demanded.68 Hence we have two equations to determine the two unknowns: the quantities

66The requirement of competition was indirect: one characteristic of a perfect market was that "there must be no conspiracies for absorbing and holding supplies to produce unnatural ratios of exchange" (Theory [4th ed.], p. 86). It is evident that the trading body could not properly be used to explain prices, because its composition depended upon prices.

67 Jevons (ibid., p. 95) stated the law of indifference as

dx2 x2 dxx1 xI

This notation is ambiguous (see Marshall, Memori- als, p. 98; F. Y. Edgeworth, Mathematical Psychics [London: Paul, i88i], pp. iio ff.).

68Jevons seems to have introduced the trading bodies to get quickly to market prices, not because of an intuition that bilateral monopoly was inde- terminate; at least he overlooked the difficulties in duopoly (Theory [4th ed.], p. II7).

UTILITY THEORY 3I9

of Xi and X2 exchanged. Quite aside from the ambiguous concept of a trading body, this procedure was illicit on his own view that utilities of different individuals are not comparable.69

Walras succeeded in establishing the correct relationship between utility and demand. He first derived the equations of maximum satisfaction for an individual: if there are m commodities, and a unit of commodity Xi is the nunmeraire in terms of which the prices of other commodities are expressed (so pi i),

we have (m - i) equations:70

MU2 MU3 P2 P3

Finally, the budget equation states the equality of values of the initial stocks of commodities (x?)and the stocks held after exchange:

X1 ? x2p2 + X3p3 + **

-I x??2P2?+X3?P3 +

We thus have mn equations to determine the m quantities of the commodities demanded or supplied by the individual. We may solve the equations for the quantities demanded or supplied as functions of the prices:

2 = X2 (P2, P3, . * X3=x3 (P2, (P2 ,

69 "The reader will find, again, that there is never, in any single instance, an attempt made to compare the amount of feeling in one mind with that in another" (ibid., p. I4).

70 Alements, Le~on 8. Let total utility -f(x1) + g(X2) + h(x3) + ... . In one of these utility functions, substitute the budget limitation,

XI + X2 P2 + X3 P3 + * * -

- 02p2 + X03P3 + *

where x4, 4, X3,. are the initial stocks. Then maximize total utility to obtain the equations in the text.

xi= (x,+xP2+xP3?...) X1 =(X10+ X0p (X30P2 + - 3P3.) -(x2p2+ x3p3+..*.).-

The xi, X2, X3 . .. , are the quantities held (demanded), and (x1-x1), (x2-x2), (X3-X3) ..A . the quantities supplied.7'

To determine the market prices, we simply add the demands of all n individuals in the market for each commodity

n n

X2= EW X2-= E X2 (p2, P3 )

n n

X3 = A X3 = E X3 (P2, P3,)

and equate the quantities demanded to the quantities available (kX7)

XO=X 2 2

XI=X 3 3

There are (in - i) such equations with which to determine the (m - i ) prices of X2, X3, ... , in terms of Xi. It may appear that we have forgotten the budget equation, but it is not an independ- ent relationship because it can be deduced from the other equations. If we multiply the last set of equations by the respective prices of the commodities and add, we obtain

t2(2X2) + p3 (X 3- X3)+...= .

But if we add the individual budget equations we obtain

xl- X>1 = P2 (X2- X2)

1* Xo_ v A3 + 0

"' This summary differs in notation and detail, but not in substance, from Walras' exposition (ibid., pp. I23 if.). The chief difference of detail is that Walras writes the utility as f(x'i?xi), where I write it asf(xi), so his xi can be negative.


Hence if the quantity demanded equals the quantity available in (m - i) mar- kets, the equality must also hold in the mth market. This is equivalent to say- ing that if we know the amounts of (m - i) commodities that have been exchanged for each other and an ninth commodity, and the rates of exchange, we necessarily know the amount of the mth commodity exchanged.

The (Walrasian) demand function is thus the relationship between the quantity of a commodity and all prices, when the individual's (or individuals') money income and tastes (utility functions) are held constant. We shall adhere to this meaning of the demand function or "curve" (the two-dimen- sional illustration of course requiring that all prices except that of the commodity are held constant), and the relationship between quantity and money income (all prices and tastes being held constant) will be designated as the income curve.

D. THE APPLICATIONS OF THE THEORY

Jevons gave only one application of his utility theory: a demonstration that both parties to an exchange gain satisfaction. The demonstration, as he gave it, was inconsistent with his denial of the possibility of comparing utilities of individuals, for it rested on the marginal utility curves of nations.72

Menger was even less specific but surely vastly more persuasive in his applications of the theory: he made it the basis of economic theory. The theory was given many everyday illustrations (mostly hypothetical, to be sure): it explained exchange, the wages of tex-

72 Theory (4th ed.), pp. 142 if. In the Preface to the second edition he proposed broader applications much closer to those of Menger and Walras but never worked out this position.

tile workers during the Civil War cot- ton shortage, the shifts of goods between free and economic, etc. More important, the theory of production became simply an instance of the theory of marginal utility: productive services were distinguished from consumption services only in being goods of higher order. Menger's version had no predic- tive value, nor did he conjecture any new economic relationships. Indeed at least two of the founders of marginal utility theory-Jevons was the exception-knew much less about economic life than a dozen predecessors such as Smith and Babbage. Yet the theory served to systematize a variety of known facts of everyday observation and seemed to confer an air of generality and structural elegance upon price theory.

XValras also did a good deal of this reorientation of economic theory in terms of utility, whereby the value of productive services was determined by the values of products. But he also attempted a specific and natural application of the theory to demand-curve analysis.

This application was the derivation of the law that price reductions will increase the quantity demanded; price increases will decrease the quantity demanded.73 Walras treated this as intui- tively obvious, but it was a strict implication of his theory. Consider the equations of maximum satisfaction:

MU lT1 M U2 M U3

71 P 2 - -

Assume p2 falls by 5p2, and assume that the individual is deprived of his nominal increase in real income, X2ap2.

At the new price, P2 - 6P2, the individ-

73 Ele'cn~ts, pp. 131, 133.

UTILITY THEORY 32I

ual obtains a larger marginal utility per dollar from X2 than from other commodities, hence he will substitute X2 for other commodities. Restore now the increment of income X23p2, and it will be used to purchase more of every commodity, including X2. The individual necessarily buys more X2 at a lower price, and therefore all individuals buy more of X2 at a lower price: the demand curve for each product must have a negative slope.74

A second application of utility theory was made in the theorem on the distribution of stocks: a redistribution of initial stocks of goods among the individuals in a market, such that each individual's holdings have the same market value before and after the redistribution, will not affect prices.75 It is the amount of income, not its composition in terms of goods, that influences consumer behavior. The most interesting point with respect to this obvious theorem is that Walras stopped here on the threshold of the analysis of the effects of income upon consumption. One may conjecture that his penchant for analyzing what are essentially barter problems in his theory of exchange played a large role in this failure to analyze income effects.76

The theory of utility also led Walras "The validity of this argument depends on the

assumption that the marginal utility of a commodity is a (diminishing) function only of the quantity of that commodity (see Sec. IV).

Ibid., pp. I45-49. 7 Perhaps mention should also be made of the

applications of utility theory to labor. Jevons' theory of disutility was labored and at times confused (see my Production and Distribution Theories [New York: Macmillan, 194I], chap. ii). Walras' treatment was more elegant-he introduced the marginal utility of leisure in complete symmetry to the theory of consumption-but not much more instructive (Elements, p. 209). Menger denied that labor was usually painful (op. cit., p. 149 n.).

to his theory of multiple equilibria.77 This theory deals with the exchange of one commodity for another in a competitive market, when both commodities have utility to the individual.78 The possessors of X1 have a fixed stock- how much will they offer at various prices of X1 (in terms of X2)? When pi is zero (no X2 is given in exchange

p11

S

Al

A'

01 XI~~~~~~~~~~~~~~~~~~~~~~~~~1

FIG. 2

for a unit of X1), they will naturally supply no Xi; the supply curve begins at (or above) the origin. At higher pi, they will offer more X1 to obtain more X2, but beyond a certain price, L, further increases in the price of X1 will lead them to reduce the quantity of X1 offered because they become relatively sated with X2. Walras illustrates this with Figure 2, where D is the demand curve and S the supply curve. A' and A" are points of stable equilibrium, be-

"Marshall's theory of multiple equilibria is in- dependent of utility analysis; it refers only to the long run, whereas Walras' theory is strictly short run. See Marshall, Pure Theory of Domestic Values ("London School Reprints" [London, I930]).

78 Elements, pp. 68-70; Wicksell restates the theory, Lectures on Political Economy (London: Macmillan, I934), I, 55 ff.


cause at higher prices the quantity supplied exceeds the quantity demanded and at lower prices the quantity demanded exceeds the quantity supplied. Point A, however, is an unstable equilibrium because at higher prices the quantity demanded exceeds the quantity supplied so the price rises even more, and conversely at lower prices. We shall not follow the history of multiple equilibria, in which economists have usually taken an apprehensive pride.

In the area of welfare economics, Walras' most important application was the theorem on maximum satisfaction:

Production in a market governed by free competition is an operation by which the [productive] services may be combined in products of appropriate kind and quantity to give the greatest possible satisfaction of needs within the limits of the double condition that each service and each product have only one price in the market, at which supply and demand are equal, and that the prices of the products are equal to their costs of production.79

This theorem, which is not true unless qualified in several respects, gave rise to an extensive literature which lies out- side our scope.80

J9Elements, p. 231 ; Jevons also stated the theorem (Theory [4th ed.], p. I41).

80 Among the important writings during our period are: A. Marshall, Principles of Economics (ist ed.; London: Macmillan, i8go), Book V, chap vii; V. Pareto, "II Massimo di utility dato dalla libera concorrenza," Giornale degli economist, Series 2, No. 9 (July, I894), pp. 48-66; E. Barone, "The Ministry of Production in the Collectivist State," reprinted in F. A. Hayek, Collectivist Eco- nomic Planning (London: Routledge, I938); K. Wicksell, Lectures on Political Economy (London: Macmillan, I934), I, 72 ff.; L. Bortke- witch, "Die Grenznutzentheorie als Grundlage einer ultra-liberalen Wirtschaftspolitik," Jahrbuch ffur Gesetzgebung, Verwaltung und Volkswirtschaft, XXII (i898), I177-I2i6; and A. C. Pigou, Wealth and Welfare (London: Macmillan, 19I2).

IV. THE FORM OF THE UTILITY

FUNCTION

The three founders of the utility theory treated the utility of a commodity as a function only of the quantity of that commodity. If Xl, X2, X3, ..., are the commodities, the individual's total utility was written (explicitly by Jevons and Walras, implicitly by Menger), as

f(xI) +g(x2) +Ih(X3) +....

They further assumed that each commodity yielded diminishing marginal utility. This form of utility function has the implication that the demand curve for each commodity has a negative slope, as I have already remarked. It has also the implication that an increase in income will lead to increased purchases of every commodity. This is easily shown with the fundamental equations,

M U1 M U2 M U3 M ,= pi P2 -_P3

If income increases, the marginal utility of every commodity (and of income) must decrease, but the marginal utility of a commodity can be reduced only by increasing its quantity. This implication was not noticed.

Edgeworth destroyed this pleasant simplicity and specificity when he wrote the total utility function as sp (Xi, X2, X3,

... ). He appears to have made this change partly because it was mathe- matically more general, partly because it was congruent with introspection.8' The change had important implications for the measurability of utility that I shall discuss in Section V.

With the additive utility function, diminishing marginal utility was a sufficient condition for convexity of the in-

81 Mathematical Psychics, pp. 20, 34, I04, I08.

UTILITY THEORY 323

difference curves; 82 with the generalized utility function, diminishing marginal utility was neither necessary nor sufficient for convex indifference curves.83 Nevertheless, Edgeworth un- necessarily continued to assume diminishing marginal utility, but he also postulated the convexity of the indifference curves. 84

Even with convexity, the generalized utility function no longer has the corollary that all income curves have positive slopes (or, therefore, that all demand curves have negative slopes).

82 Diminishing marginal utility for each commodity was not necessary, however: the indifference curves could be convex to the origin if every commodity except one yielded diminishing marginal utility, and the marginal utility of this exception commodity did not increase too rapidly. This exceptional case was first analyzed by Slutsky (see Sec. VII).

In the two-commodity case

dxl 2

is the slope of an indifference curve, and the condition for convexity is

d2x, 2 -2 d22 (2(P11-2 P1(P2(12 + (PI22 9 3 - --~~~> 0

where the subscripts to so denote partial differen- tiation with respect to the indicated variables. It is clear that diminishing marginal utility (spjj and C722 negative) is not necessary for convexity, since

S?12 can be positive and large, and it is not sufficient, since IP12 can be negative and large. In the additive case ( P12 = 0), at most one marginal utility can be increasing, as was pointed out in the previous footnote.

84 Mathematical Psychics, p. 36. He wrote the tility function as so(x1,-x2), in my notationu, for reasons which will be pointed out below. He postulated that P12 <O, where -X2 is work done by the person and X1 is remuneration received. This is equivalent to assuming that an increase in remuneration increases the marginal utility of leisure, and would be represented by f012> 0 if we write the function as (P(x1, x2), as is now customary. With diminishing marginal utility this condition leads to convexity (see previous note).

After a price reduction, 8P2, we may again segregate the effect of a change in relative prices by temporarily reducing the individual's income by x28p2. When we restore this increment of real income, we cannot be sure that each commodity will be consumed in larger quantity. Suppose an increase in Xi reduces the marginal utility of X2. Then when a portion of the increment of real income x28p2 is spent on X1, MU2 may diminish so much that the amount of X2 must be reduced below its original quantity to fulfil the maximum satisfaction conditions.85

The only further generalization of the utility function (aside from questions of measurability) was the inclu- sion of the quantities consumed by other people in the utility function of

'5The conditions for maximum satisfaction are

Oi Pi

SP2 P2'

xjp 1-- X2p2= R.

Differentiate these equations with respect to R (holding prices constant) and solve to obtain

Ox2 P2fs1 - PI'P12

OR p22011 - 2P1P2sP2 t1- pj S22

The denominator of the right side is negative if the indifference curves are convex to the origin. The numerator, however, can be positive with 'p <0, so the whole expression may be negative (X2 may be "inferior"). With the additive function, 0p12= (and of course they assumed (pii <0), so the expression must be positive (X2 [and X1] must be "normal"). Similarly, differentiate the equations with respect to P2 holding Pi and R constant) and solve to obtain

aX2 P]I< + X2P I2 -X2p211

OIP P22] - 2PIP2f01 + - 22P ,

Again the denominator is negative, and the numerator may be negative if (P12 is negative, so the whole expression may be positive. With the additive utility function and diminishing marginal utility, the expression must be negative.


the individual. Thus one's pleasure from diamonds is reduced if many other people have them (or if none do!), and one's pleasure from a given income is reduced if others' incomes rise. This line of thought is very old,"6 but it was first introduced explicitly into utility analysis in I892. Fisher casually suggested it:

Again we could treat [utility] as a function of the quantities of each commodity produced or consumed by all persons in the market. This becomes important when we consider a man in relation to the members of his family or consider articles of fashion as diamonds, also when we account for that (never thoroughly studied) interdependence, the division of labor.87

Henry Cunynghame made the same suggestion more emphatically in the same year:

Almost the whole value of strawberries in March, to those who like this tasteless mode of ostentation, is the fact that others cannot get them. As my landlady once remarked, "Surely, sir, you would not like anything so common and cheap as a fresh herring?" The demand for diamonds, rubies, and sapphires is another example of this.88

Pigou took up this argument, used it to show that consumer surpluses of various individuals cannot be added, but decided that these interrelation- ships of individuals' utilities were stable (and hence did not vitiate the consumer surplus apparatus) when the price changes were small.89 It was only proper that Marshall's leading pupil

"E.g.: A. Smith, Theory of Moral Sentiments (Boston: Wells & Lilly, i817), Part III, chap. iii; Part IV, chap. i; N. F. Canard, Principes d'economie politique (Paris: Buisson, i8oi), chap. v; Senior, op. cit., p. 12.

" Mathematical Investigations in the Theory of Value and Prices (New Haven: Yale University Press, 1937-reprint of i892 ed.), p. 102. Fisher independently reached the generalized utility function of Edgeworth (ibid., Preface).

88 "Some Improvements in Simple Geometrical Methods of Treating Exchange Value, Monopoly, and Rent," Economic Journal, II (i892), 37.

should postulate the constancy of the marginal utility of prestige.

Pigou's article elicited the first sta- tistical investigation designed to test a utility theory (and apparently the only such investigation during the period). Edgeworth, a Fellow of All Souls, col- lected statistics from "a certain Oxford College" to determine "whether the size of the party has any influence upon the depth of the potations"-that is, upon the per capita consumption of wine. The data were presented in relative form lest they "should excite the envy of some and the contempt of others"; the conclusion was that the effect of the size of party was inappreciable."

A few subsequent attempts have been made to revive this extension of the utility function to include the effect on one person's utility of other people's consumption, but the main tradition has ignored the extension. This neglect seems to have stemmed partly from a belief in the unimportance of the effect and partly from the obstacles it would put in the way of drawing specific infer- ences from utility analysis.

There remain three subordinate topics that may conveniently be discussed here. They are (a) the graphical exposition of the theory of the generalized utility function; (b) the attitude of contemporary economists toward Edge- worth's generalization; and (c) the Bernoulli hypothesis on the shape of the utility function.

89 "Some Remarks on Utility," Economic Journal, XIII (1903), 6o if. He wrote the utility function of the individual as

U =4 [x, y, z, w, K (ab)], where x, y, z, and w were quantities consumed by the individual ,ai was the quantity of x possessed by some other individual i, whose social distance was bi, and K was a symbol "akin to, though not identical with, the ordinary I" (ibid., p. 6i).

90 Papers Relating to Political Economy (London: Macmillan, 1925), II, 323-24n.

UTILITY THEORY 325

A. INDIFFERENCE CURVES

With the introduction of the inter- relationship of utilities of commodities, it was no longer possible to portray total utility graphically in two dimensions. Edgeworth devised indifference curves, or contour lines, to permit of a graphical analysis of utility in this case. In itself this was merely an expositional advance, but it merits summarization because of its great popularity in modern times and because it later invited attention to questions relating to the measurability of utility.

We restrict ourselves to the case of two commodities, as Edgeworth and almost everyone since has done in graphical analysis.91 We define the indifference curve as the combinations of Xi and X2 yielding equal satisfaction, i.e., sP(X1,X2) constant. Edgeworth chose an asymmetrical graphical illustration of these curves that had a defi- nite advantage for his purpose of analyzing bilateral monopoly. He let the abscissa represent the quantity of X1 obtained by the individual, and the ordinate represent the quantity of X2 given up.

It is evident that such indifference curves have a positive slope (if both commodities are desirable), for the individual will require more Xi to offset (in utility) the loss of more X2. In fact, the slope of the indifference curve with respect to the X1 axis will be

dX2 M U1 92

dx1 MU2 In addition, Edgeworth postulated that

91 The three commodity indifference surfaces are of course the limit of literal graphical exposition, and even they have been deemed unappetizingly complex.

92 For dx1MU, will be the gain of utility from an increment dxi, and dx2MU2 will be the loss of utility from a decrement dx2, and these must be equal if the movement is along an indifference curve.

the indifference curves are concave to the Xi axis.

Edgeworth's pioneer demonstration of the indeterminacy of bilateral monopoly will illustrate the advantage of this formulation.3 A trader possessing X2 but no X1 would be at the origin; his indifference curves are those labeled I in Figure 3. The second trader, who

X2

11 112

10

10

..111~~~~1

FIG. 3

possesses X1 but no X2, will have the corresponding indifference curves (II), for he will be giving up X1 and acquir- ing X2 in exchange. The points where the two sets of indifference curves are tangent form a curve, CC, which Edge- worth christened the contract curve. The ends of the contract curve are detern-ined by the condition that no trader be worse off after trading than before, i.e., by the indifference curves, Io and 11o. The final contract between the traders must take place on this contract curve, because if it occurred elsewhere, it would be to the gain of one party, and not to the loss of the other, to move to the curve. Thus point Q was not a tenable point of final contract because individual II can move from

'Mathematical Psychics, pp. 20 ff.


II, to the higher indifference curve II2, while I remains on the same indifference curve, I,. Any point on the contract curve is a position of possible equilibrium, and the precise position reached will be governed by gigglingg dodges and designing obstinacy, and other incalculable and often disreputa- ble accidents."94

Although this mode of exposition is convenient in the analysis of trade in two commodities between two individuals, it has no special advantage in the competitive case, and asymmetrical axes are awkward in algebraic analysis. Fisher introduced the now conven- tional graphical statement, in which the amounts held (or obtained) of the commodities appear on all axes.5

B. CONTEMPORARY PRACTICE

Despite the intuitive appeal of Edge- worth's generalized utility function, economists adhered to the additive utility function with considerable tenacity. In the nonmathematical writings, such as those of Bo3hm-Bawerk, Wieser, and J. B. Clark, the additive function was used almost exclusively. Barone defended it as an approximation.6 Wick- sell used it exclusively in his Uber Wert (1894), although conceding the greater realism of the generalized function,97 and found some place for it in his later Lectures.8 Wicksteed used only the additive function in his Alphabet (I888)f9

and also in the elementary exposition of the theory in his Common Sense (i9io) but not in the "advanced" statement.'00 Finally, Marshall and Pareto were so influential as to require more ex- tended discussion.

Marshall also started with the Jev- " Ibid., p. 46. I" Op. cit., Part II. " Le Opere economniche (Bologna: Zanichelli,

I936), I, esp. pp. 22-23.

ons-Walras assumption, to which he had probably arrived independently. This assumption was not explicit in the first edition of the Principles (i890),

but one can cite evidence of its pres- ence.

First, in his mathematical characteri- zation of the utility function Marshall ignores any interdependence of utilities.10' Second, he asserts the law of negatively sloping demand curves in all generality: "There is then one law and only one law which is common to all demand schedules, viz. that the greater the amount to be sold the smaller will be the price at which it will find purchasers."'102 This is a corollary of diminishing marginal utility only if the utility function is additive. Third, he was prepared to measure the utility of all commodities as the sum of the individual utilities: "We may regard the aggregate of the money measures of the total utility of wealth as a fair measure of that part of happiness which is dependent on wealth."''03

In the second edition (I89I) the as-

9' Uber Wert, Kapital und Rente (Jena: Fischer, I894), esp. p. 43.

98 Lectures on Political Economy, I, 46-47, 55 ff.; however, the generalized function is preferred (ibid., pp. 4I-42, 48-49, 79 if) .

9 Alphabet of Economic Science (London: Mac- millan, i888).

"1 Common Sense of Political Economy (London: Routledge, I934), Vol. I, chap. ii; Vol. II, chap. ii; the generalized function is used in Vol. II, chap. iii, esp. p. 479.

101 Principles of Economics (London: Macmillan, i890), Mathematical Notes II, III, VII [I, II, VI]. References in brackets will be used for corresponding passages in the eighth edition.

102 Ibid., pp. I59-60 [991. 103 Ibid., pp. I79-8o, also Mathematical Note VII.

His Mathematical Note III [II] also implies an additive function if his p, "the price which [a person] is just willing to pay for an amount [x] of the commodity . . ." is interpreted as our x1po and the price to the person is treated as constant. See Sec. VII.

UTILITY THEORY 3 27

sumption became reasonably explicit: Prof. Edgeworth's plan of representing U and

V as general functions of x and y has great at- tractions to the mathematician; but it seems less adapted to express the everyday facts of economic life than that of regarding, as Jevons did, the marginal utilities of apples as functions of X simply.'04

The facts both of everyday life and of contemporary theory soon led Marshall to make serious qualifications of his theory but never to qualify this statement.

Even in the first edition Marshall had inconsistently recognized the existence of "rival" products, which were defined as products able to satisfy the same desires.'05 Fisher's discussion of competing and completing goods seems to have been the stimulus to Marshall to give more weight to interrelation- ships of utilities in the third edition of the Principles (I895) .106 Once persuaded, Marshall modified his theory on two points. The first was that he slightly modified his assertion of the universality of negatively sloping demand curves and in fact introduced the Giffen paradox as an exception.'07 The second alteration was in his treatment of consumers' surplus: "When the total utilities of two commodities which con- tribute to the same purpose are calcu- lated on this plan, we cannot say that the total utility of the two together is equal to the sum of the total utilities of each separately."'108 No important changes were made thereafter.

104 Loc. cit., p. 756 [845]. See also the deduction of diminishing marginal utility from negatively sloping demand curves (ibid., p. 159 [ioi n.]).

101 See Sec. VI.

106 Reference is there made to Fisher's "brilliant" book, precisely on this point (Principles [3d ed.; London: Macmillan, 1895], p. 460 n. [39o n.]). For Fisher's discussion see Sec. VI below.

"@Loc. cit., p. 208 [I32]. See my "Notes on the History of the Giffen Paradox," Journal of Political Economy, LV ('947), I52-56.

These alterations were only patch- work repairs; Marshall did not rework his theory of utility. He retained to the last a theory constructed on the assumption of an additive utility function.

Pareto also conceded the validity of the Edgeworth generalization but continued to use chiefly the additive function in his early work.'09 Indeed, he offered the remarkable argument:

One sees now that instead of being able to use the indicated properties of the final degree of utility to demonstrate what laws demand and supply must obey, it is necessary to follow the opposite path, and use the knowledge of such laws one may obtain from experience to derive the properties of the final degree of utility. One cannot rigorously demonstrate the law of demand, but rather, from the directly observable fact that demand diminishes with the increase of price we deduce the consequence that the final degrees of utility may each be considered -as far as this phenomenon is concerned-as approximately dependent only on the quantity of the commodity to which it is related.1'0

In the Manuel, however, he showed that the additive utility function leads to conclusions which are contradicted by experience,"' but defended it as an approximation which was permissible for large categories of expenditure and for small changes in the quantities of substitutes or complements."2 There is no reason to believe that this is true.

[To be concluded]

108 He added the less than candid footnote: "Some ambiguous phrases in earlier editions appear to have suggested to some readers the opposite opinion" (lc. cit., p. 207 and n. [3I and n.]).

10 "Considerazioni sui principii fondamentali dell'economia politica pura," Giornale degli econo- misti, Series 2, Vol. V (August, i892); Cours d'Jco- noinie politique (Lausanne: Rouge, i897), IT, 332 ff.

'0 "Considerazioni . . . ," op. cit., VII (I893), 307. . Below, Sec. VII. 12 Manuel d'e'conomie politique (2d ed.; Paris:

Giard, I927), pp. 253 ff., 274.

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